The built-in solution to change the order of posts in a category is to change the “Order” attribute of each post. If your post type is set to support “Post Attributes”, you should have a box that looks like this in your right sidebar when editing the post:
0 will be the first in the list, and ordered in ascending numerical value. Note that this will change the order of wherever your posts are displayed on your website, not specific to a single category page.